We study the problem of learning unknown parameters in stochastic interacting particle systems with polynomial drift, interaction and diffusion functions from the path of one single particle in the system. Our estimator is obtained by solving a linear system which is constructed by imposing appropriate conditions on the moments of the invariant distribution of the mean field limit and on the quadratic variation of the process. Our approach is easy to implement as it only requires the approximation of the moments via the ergodic theorem and the solution of a low-dimensional linear system. Moreover, we prove that our estimator is asymptotically unbiased in the limits of infinite data and infinite number of particles (mean field limit). In addition, we present several numerical experiments that validate the theoretical analysis and show the effectiveness of our methodology to accurately infer parameters in systems of interacting particles.

翻译：我们研究在随机交互粒子系统中，通过系统中单个粒子的轨迹，学习具有多项式漂移、交互和扩散函数的未知参数问题。该估计量通过求解一个线性系统获得，该系统基于对平均场极限不变分布矩以及过程二次变分施加适当条件而构建。该方法易于实现，仅需通过遍历定理近似矩并求解低维线性系统即可。此外，我们证明在无限数据极限和无限粒子数极限（平均场极限）下，该估计量渐近无偏。最后，通过若干数值实验验证理论分析，并展示该方法在精确推断交互粒子系统参数方面的有效性。